Intro
Models
I wanted to test for an eventual effect of logging through light and disturbance on Symphonia individuals growth. I used control and treatment plots between 1988 and 1992. I kept only trees already present in 1988 and still alive in 1992, and calculate their growth during this time. I then tried to look at the effect of both distance to the closest logging gaps (\(d_{gaps}\) in \(m\)), original dbh of inidividuals in 1998 (\(dbh_{1998}\) in \(cm\)), and their interaction on growth (\(growth\) in \(cm\)). I used a bayesian model following a log normal law for growth : \[M_{log}:~~growth \sim log \mathcal{N}(\alpha*log(d_{gaps}+1) + \beta*dbh_{1998} + \gamma*log(d_{gaps}+1)*dbh_{1998}, \sigma) \]
Then I used the model proposed by Herault et al. (2010): \[growth_i \sim \mathcal{N}(\mu*\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta);\sigma)~|~i\in[1:I]~,~j\in[1:J]\] where the growth between 1988 and 1992 (\(growth\) in \(cm\)) of an individual \(i\) depend on its distance to logging gaps \(j\) (\(d_{i,j}\) in \(m\)) and the logging gap surface (\(S_j\) in \(m\)). \(\mu\), \(\alpha\), and \(\beta\) represents are the disturbance parameters. A nul \(\alpha\) or \(\beta\) indicates a nul effect of logging gaps distance or surface. That i declined in different versions :
Table 1: Models summary.
| M | Model |
|---|---|
| \(M_{\mu, \alpha, \beta}\) | \(growth_i \sim \mathcal{N}(\mu*\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta);\sigma)\) |
| \(M_{\alpha, \beta}\) | \(growth_i \sim \mathcal{N}(\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta);\sigma)\) |
| \(M_{\mu, \alpha, \beta, \omega}\) | \(growth_i \sim \mathcal{N}(\mu*\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta)+\omega;\sigma)\) |
Map
Map of Symphonia growth after logging (between 1988 and 1992) and distance to logging gaps. dots size and label represents the growth in cm between 1988 and 1992
\(M_{log}\)
Column 1
The model seems to have correctly converged besides log likelyhood seems to be constrained to a maximal value (see markov chain plot), and parameters seems not much correlated besides a small link between \(\beta\) and \(\gamma\) (see parameters pairs plot). So if we consider the model as valid, parameter posterior distribution seems to indicate a strong effect for the logarithm distance to the gap (\(\alpha\) distribution don’t overlap 0 with a mean at \(\alpha_m = 0.18\)). The growth of Symphonia individuals being increased close to gaps by almost 14% (\(e^{\alpha_m}=1.14\)).
Model Predictions.
Parameters markov chains.
Column 2
\[growth = e^{0.13*log(d_{gaps}+1) + 0.03*dbh_{1988} + -0.01*log(d_{gaps}+1)*dbh_{1988}} \]
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{\mu, \alpha, \beta}\)
Column 1
Model Predictions.
Parameters markov chains.
Column 2
\(M_{\mu, \alpha, \beta}\): \(growth_i \sim \mathcal{N}(\mu*\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta);\sigma)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{\alpha, \beta}\)
Column 1
Parameters markov chains.
Column 2
\(M_{\alpha, \beta}\): \(growth_i \sim \mathcal{N}(\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta);\sigma)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
\(M_{\mu, \alpha, \beta, \omega}\)
Column 1
Parameters markov chains.
Column 2
\(M_{\mu, \alpha, \beta, \omega}\): \(growth_i \sim \mathcal{N}(\mu*\sum_j^J(e^{-\alpha*d_{i,j}}*S_j^\beta)+\omega;\sigma)\)
Parameters posterior ditribution. Light blue area represents 80% confidence interval, and vertical blue line the mean.
Parameters pairs plot. Parameters density distribution, pairs plot and Pearson’s coefficient of correlation.
References
Herault, B., Ouallet, J., Blanc, L., Wagner, F. & Baraloto, C. (2010). Growth responses of neotropical trees to logging gaps. Journal of Applied Ecology, 47, 821–831.